Uplink signal transmission and reception using optimized rank 3 codebook

ABSTRACT

A method for transmitting and receiving uplink signals using an optimized rank  3  codebook is disclosed. The optimized rank 3 codebook includes 6 precoding matrix groups, each of which has 1 variable having an amplitude of 1. Preferably, the optimized 4Tx rank  3  codebook has 12 precoding matrix, two precoding matrixes are selected from each the above 6 precoding matrix groups considering chordal distance and the number of precoding matrix.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/836,433, filed on Aug. 26, 2015, now U.S. Pat. No. 9,294,175, whichis a continuation of U.S. patent application Ser. No. 14/645,834, filedon Mar. 12, 2015, now U.S. Pat. No. 9,154,343, which is a continuationof U.S. application Ser. No. 13/254,161, filed on Aug. 31, 2011, nowU.S. Pat. No. 9,007,885, which is the National Stage filing under 35U.S.C. 371 of International Application No. PCT/KR2010/002261, filed onApr. 13, 2010, which claims the benefit of earlier filing date and rightof priority to Korean Patent Application No. 10-2010-0031369, filed onApr. 6, 2010, and also claims the benefit of U.S. ProvisionalApplication Nos. 61/219,396, filed on Jun. 23, 2009, 61/186,386, filedon Jun. 12, 2009, 61/185,198, filed on Jun. 9, 2009, 61/184,830, filedon Jun. 7, 2009, 61/184,313, filed on Jun. 5, 2009, 61/183,065, filed onJun. 1, 2009, 61/181,666, filed on May 28, 2009, and 61/168,931, filedon Apr. 13, 2009, the contents of which are all hereby incorporated byreference herein in their entirety.

BACKGROUND OF THE INVENTION

Field of the Invention

The present invention relates to a wireless mobile communication system,and more particularly to a communication system based on a MultipleInput Multiple Output (MIMO) scheme.

Discussion of the Related Art

MIMO technology is an abbreviation for Multiple Input Multiple Outputtechnology. MIMO technology uses a plurality of transmission (Tx)antennas and a plurality of reception (Rx) antennas to improve theefficiency of transmission and reception (Tx/Rx) of data. In otherwords, MIMO technology allows a transmission end or reception end of awireless communication system to use multiple antennas (hereinafterreferred to as a multi-antenna), so that the capacity or performance canbe improved. For convenience of description, the term “MIMO” can also beconsidered to be a multi-antenna technology.

In more detail, MIMO technology is not dependent on a single antennapath to receive a message. Instead, the MIMO technology collects aplurality of data fragments received via several antennas, merges thecollected data fragments, and completes total data. As a result, MIMOtechnology can increase a data transmission rate within apredetermined-sized cell region, or can increase system coverage whileguaranteeing a spec specific data transmission rate. Under thissituation, MIMO technology can be widely applied to mobile communicationterminals, repeaters, or the like. MIMO technology can extend the rangeof data communication, so that it can overcome the limited amount oftransmission data of mobile communication systems.

FIG. 1 is a block diagram illustrating a general MIMO communicationsystem.

Referring to FIG. 1, the number of transmission (Tx) antennas in atransmitter is N_(T), and the number of reception (Rx) antennas in areceiver is N_(R). In this way, theoretical channel transmissioncapacity of the MIMO communication system when both the transmitter andthe receiver use a plurality of antennas is greater than that of anothercase in which only the transmitter or the receiver uses severalantennas. The theoretical channel transmission capacity of the MIMOcommunication system increases in proportion to the number of antennas.Therefore, data transmission rate and frequency efficiency are greatlyincreased. Provided that a maximum data transmission rate acquired whena single antenna is used is set to R_(o), data transmission rateacquired when multiple antennas are used can theoretically increase by apredetermined amount that corresponds to the single antenna datatransmission rate (R_(o)) multiplied by a rate of increase R_(i). Therate of increase (R_(i)) can be represented by the following equation 1.R _(i)=min(N _(T) , N _(R))  [Equation 1]

For example, provided that a MIMO system uses four transmission (Tx)antennas and four reception (Rx) antennas, the MIMO system cantheoretically acquire a high data transmission rate which is four timeshigher than that of a single antenna system. After the above-mentionedtheoretical capacity increase of the MIMO system was demonstrated in themid-1990s, many developers began to conduct intensive research into avariety of technologies which can substantially increase a datatransmission rate using the theoretical capacity increase. Some of theabove technologies have been reflected in a variety of wirelesscommunication standards, for example, a third-generation mobilecommunication or a next-generation wireless LAN, etc. Theabove-mentioned MIMO technology can be classified into a spatialdiversity scheme (also called a Transmit Diversity scheme) and a spatialmultiplexing scheme. The spatial diversity scheme increases transmissionreliability using symbols passing various channel paths. The spatialmultiplexing scheme simultaneously transmits a plurality of data symbolsvia a plurality of transmission (Tx) antennas, so that it increases atransmission rate of data. In addition, the combination of the spatialdiversity scheme and the spatial multiplexing scheme has also beenrecently developed to properly acquire unique advantages of the twoschemes.

In association with the MIMO technology, a variety of MIMO-associatedtechnologies have been intensively researched by many companies ordevelopers, for example, research into an information theory associatedwith a MIMO communication capacity calculation under various channelenvironments or multiple access environments, research into radiofrequency (RF) channel measurement and modeling of the MIMO system, andresearch into a space-time signal processing technology for increasingtransmission reliability and data transmission rate.

In a 3^(rd) Generation Partnership Project Long Term Evolution (3GPPLTE) system, the above-mentioned MIMO scheme is applied to only downlinksignal transmission of the 3GPP LTE system. The MIMO technology may alsohe extended to uplink signal transmission. In this case, a transmitterstructure should be changed to implement the MIMO, technology, so that aPeak power to Average Power Ratio (PAPR) or Cubic Metric (CM)characteristics may be deteriorated. Therefore, there is needed a newtechnology capable of effectively applying the MIMO scheme to uplinksignal transmission.

Specifically, an appropriate number of precoding matrices are selectedfrom a codebook for use in uplink rank 3 transmission, and at the sametime it is necessary for the codebook to maximize a chordal distancebetween precoding matrices.

SUMMARY OF THE INVENTION

Accordingly, the present invention is directed to an uplink signaltransmission and method using an optimized rank 3 codebook thatsubstantially obviates one or more problems due to limitations anddisadvantages of the related art.

Additional advantages, objects, and features of the invention will beset forth in part in the description which follows and in part willbecome apparent to those having ordinary skill in the art uponexamination of the following or may be learned from practice of theinvention. The objectives and other advantages of the invention may berealized and attained by the structure particularly pointed out in thewritten description and claims hereof as well as the appended drawings.

To achieve these objects and other advantages and in accordance with thepurpose of the invention, as embodied and broadly described herein, amethod for controlling a user equipment (UE) to transmit uplink signalsvia four antennas includes mapping the uplink signals to three layers;performing Discrete Fourier Transform (DFT) spreading upon each ofsignals of the three layers, precoding the three DFT-spread layersignals using a specific precoding matrix selected from among aprestored codebook, performing a predetermined process for constructinga Single Carrier-Frequency Division Multiple Access (SC-FDMA) symbolupon the precoded signals, and transmitting the SC-FDMA symbol to a NodeB, i.e. Node B via the four antennas, wherein the prestored codebookconsists of 6 precoding matrix groups

$\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},{{and}\mspace{14mu}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}}$(where X is a complex variable having an amplitude of 1)). To each ofthe precoding matrixes in the codebook a predetermined constant can bemultiplied. However, this constant is not discussed in this document forconvenience.

The prestored codebook may include two precoding matrices selected fromeach of the 6 precoding matrix groups. The prestored codebook mayinclude two precoding matrices selected from each of the 6 precodingmatrix groups, wherein X values in each of the 6 precoding matrix groupshave a phase difference of 180° in a complex plane. The prestoredcodebook may include one precoding matrix having the X value of 1 eachof the 6 precoding matrix groups and the other precoding matrix havingthe X value of −1 in each of the 6 precoding matrix groups.

The prestored codebook may include the following 12 precoding matricesof

$\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},{\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}.}$

In another aspect of the present invention, a user equipment (UE) fortransmitting uplink signals via multiple antennas includes four antennasfor transmitting and receiving signals, a memory for storing a codebookused for transmitting three layer signals via the four antennas, and aprocessor connected to the multiple antennas and the memory so as toprocess transmission of the uplink signals, wherein the processorincludes a layer rapper for mapping the uplink signals to the threelayers, a Discrete Fourier Transform (DFT) module for performing DFTspreading upon each of the three layer signals, a precoder for precodingthe three DFT-spread layer signals received from the DFT module using aspecific precoding matrix selected from among the codebook stored in thememory, and transmission module for performing a predetermined processfor constructing a Single Carrier-Frequency Division Multiple Access(SC-FDMA) symbol upon the precoded signals, and transmitting theprocessed signals to a Node BNode B via the four antennas, wherein theprestored codebook consists of 6 precoding matrix groups of

$\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},{{and}\mspace{14mu}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}}$(where X is a complex variable having an amplitude of 1).

The memory may store, as the codebook, two precoding matrices from eachof the 6 precoding matrix groups. The memory may store, as the codebook,two precoding matrices in which X values in each of the 6 precodingmatrix groups have a phase difference of 180° in a complex plane. Thememory may store, as the codebook, both one precoding matrix having theX value of 1 in each of the 6 precoding matrix groups and the otherprecoding matrix having the X value of −1 in each of the 6 precodingmatrix groups.

The memory may store, as the codebook, 12 precoding matrices of

$\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},{\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}.}$

In another aspect of the present invention, method for controlling aNode BNode B to receive uplink signals of a user equipment (UE) includesreceiving signals having been transmitted via four antennas and threelayers from the user equipment (UE), and processing the received signalsusing a specific precoding matrix selected from among a prestoredcodebook, wherein the prestored codebook consists of 6 precoding matrixgroups of

$\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},{{and}\mspace{14mu}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}}$(where X is complex variable having an amplitude of 1).

The prestored codebook may consists of 12 precoding matrices of

$\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},{\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}.}$

In another aspect of the present invention, a Node B for receivinguplink signals of a user equipment (UE) includes one or multipleantennas for transmitting and receiving signals, a memory for storing acodebook used for receiving three layer signals transmitted by the UEvia four antennas of the UE, and a processor connected to the antennasand the memory so as to process reception of the uplink signals, whereinthe processor is configured to process the received signals using aspecific precoding matrix selected from among the codebook stored in thememory, wherein the codebook consists of 6 precoding matrix groups of

$\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},{{and}\mspace{14mu}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}}$(where X is a complex variable having an amplitude of 1).

The memory may store, as the codebook, 12 precoding matrices of

$\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},{\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}.}$

As described above, a method for transmitting and receiving uplinksignals according to the embodiments of the present invention maytransmit and receive using the above-mentioned optimized 4Tx rank 3codebook, transmit uplink signals by reflecting a channel situationusing a small amount of signaling overhead in various ways, and solve aPAPR problem. It is to be understood that both the foregoing generaldescription and the following detailed description of the presentinvention are exemplary and explanatory and are intended to providefurther explanation of the invention as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are included to provide a furtherunderstanding of the invention and are incorporated in and constitute apart of this application, illustrate embodiment(s) of the invention andtogether with the description serve to explain the principle of theinvention. In the drawings:

FIG. 1 is a conceptual diagram illustrating a general MIMO communicationsystem.

FIG. 2 illustrates a general structure of a transmitter based on MIMOtechnology.

FIG. 3 is a detailed diagram illustrating the general structure shown inFIG. 2.

FIG. 4 is a conceptual diagram illustrating a method for precodinginformation of each layer and transmitting the precoded information viaan antenna.

FIG. 5 is a conceptual diagram illustrating a general SC-FDMA scheme.

FIG. 6 is a conceptual diagram illustrating a method for mapping acodeword to several layers.

FIG. 7 is a conceptual diagram illustrating a method for performing DFTupon each layer after performing codeword-to-layer mapping (i.e.,codeword-layer mapping) so as to prevent a CM value for each antennafrom being increased.

FIG. 8 is a conceptual diagram illustrating a chordal distance.

FIG. 9 illustrates the relationship of chordal distances among 6precoding matrix groups according to one embodiment of the presentinvention.

FIG. 10 illustrates conditions of variables for providing a maximumchordal distance within the same precoding matrix group.

FIG. 11 illustrates the relationship of chordal distances when avariable of a precoding matrix contained in the same precoding matrixgroup includes a Quadrature Phase Shift Keying (QPSK) alphabet.

FIG. 12 is a block diagram illustrating a general Node B and a generaluser equipment (UE).

FIGS. 13 to 15 illustrate an SC-FDMA scheme for transmitting an uplinksignal in a 3GPP LTE system and an OFDMA scheme for transmitting adownlink signal in the 3GPP LTE system.

FIG. 16 illustrates a processor of a UE according to one embodiment ofthe present invention.

DETAILED DESCRIPTION OF THE INVENTION

Reference will now be made in detail to the preferred embodiments of thepresent invention, examples of which are illustrated in the accompanyingdrawings. Wherever possible, the same reference numbers will be usedthroughout the drawings to refer to the same or like parts.

The detailed description, which will be given below with reference tothe accompanying drawings, is intended to explain exemplary embodimentsof the present invention, rather than to show the only embodiments thatcan be implemented according to the present invention. The followingdetailed description includes specific details in order to provide athorough understanding of the present invention. However, it will beapparent to those skilled in the art that the present invention may bepracticed without such specific details. For example, the followingdescription will be given centering on specific terms, but the presentinvention is not limited thereto and any other terms may be used torepresent the same meanings. Also, wherever possible, the same referencenumbers will be used throughout the drawings to refer to the same orlike parts.

Peak power to Average Power Ratio (PAPR) is a parameter indicatingcharacteristics of a waveform. PAPR is specific value acquired when apeak amplitude of the waveform is divided by a time-averaged Root MeanSquare (RMS) value of the waveform. PAPR is a dimensionless value. Ingeneral, a PAPR of a single carrier signal is better than that of amulti-carrier signal. Cubic Metric (CM) is another specific value todescribe a waveform characteristic which is similar to PAPR.

An LTE-Advanced scheme can implement MIMO technology using SingleCarrier-Frequency Division Multiple Access (SC-FDMA) so as to maintain asuperior CM Property. When using general precoding, signal includinginformation corresponding to several layers is multiplexed andtransmitted via a single antenna, so that the signal transmitted viathis antenna may be considered to be a kind of multi-carrier signal.PAPR is associated with dynamic range that must be supported by a poweramplifier of a transmitter, and a CM value is another value capable ofbeing used as a substitute for the PAPR.

FIG. 2 shows a general structure of a transmitter based on MIMOtechnology.

In FIG. 2, one or more codewords are mapped to a plurality layers. Inthis case, mapping information is mapped to each physical antenna by aprecoding process, and is then transmitted via each physical antenna.

FIG. 3 is a detailed block diagram illustrating the MIMO-basedtransmitter shown in FIG. 2.

The term ‘codeword’ indicates that Cyclic Redundancy Check (CRC) bitsare attached to data information and are then encoded by a specificcoding method. There are a variety of coding methods, for example, aturbo code, a tail biting convolution code, and the like. Each codewordis mapped to one or more layers (i.e., one or more virtual layers), anda total number of mapped layers is equal to a rank value. In otherwords, if a transmission rank is 3, a total number of transmissionlayers is also set to 3. Information mapped to each layer is precoded.In this case, data information mapped to each layer is mapped to aphysical layer through a precoding process (where, the term ‘layer’means a virtual layer as far as it especially designates a physicallayer). Information is transmitted to each antenna via each physicallayer. Under the condition that no specified explanation is shown inFIG. 3, the precoding is carried out in a frequency domain, and an OFDMinformation transmission scheme is used for information mapped to thephysical laver. The information mapped to the physical layer is mappedto a specific frequency domain, and is then IFFT-processed. After that,a cyclic Prefix (CP) is attached to the IFFT result. Thereafter,information is transmitted to each antenna via a radio frequency (RF)chain.

The precoding process may be carried out by matrix multiplication. Ineach of the matrices, the number of rows is equal to the number ofphysical layers (i.e., the number of antennas), and the number ofcolumns is equal to a rank value. The rank value is equal to the numberof layers, so that the number of columns is equal to the number oflayers. Referring to the following equation 2, information mapped to alayer (i.e., a virtual layer) is x₁ and x₂, each element p_(ij) of a(4×2) matrix is a weight used for precoding. y₁, y₂, y₃, and y₄ areinformation mapped to physical layers, and are transmitted viarespective antennas using individual OFDM transmission schemes.

$\begin{matrix}{\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4\;}\end{bmatrix} = {\begin{bmatrix}p_{11} & p_{21} \\p_{12} & p_{22} \\p_{13} & p_{23} \\p_{14} & p_{24}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

In the following description, a virtual layer will hereinafter bereferred to as a layer so long as such use will not lead to confusion.An operation for mapping a virtual layer signal to a physical layer willhereinafter be considered to be an operation for directly mapping alayer to an antenna.

The precoding method can be mainly classified into two methods, i.e., awideband precoding method and a subband precoding method.

The wideband precoding method is as follows. According to the widebandprecoding method, when precoding is carried out in a frequency domain,the same precoding matrix is applied to all information transmitted tothe frequency domain.

FIG. 4 is a conceptual diagram illustrating a method for precodinginformation of each layer and transmitting the precoded information viaan antenna.

Referring to FIG. 4, it can be recognized that information correspondingto a plurality of layers is precoded while being classified according tosubcarriers of each frequency domain, and the precoded information istransmitted via each antenna. All precoding matrices in the wideband ‘P’in the wideband precoding method are equal to each other.

The subband precoding method is provided by the extension of thewideband precoding method. The subband precoding method applies avariety of precoding matrices to each subcarrier without applying thesame precoding matrix to all subcarriers. In other words, according tothe subband precoding method, a precoding matrix ‘P’ is used in aspecific subcarrier, and another precoding matrix ‘M’ is used in theremaining subcarriers other than the specific subcarrier. Herein,element values of the precoding matrix ‘P’ are different from those ofthe other precoding matrix ‘M’.

Uplink signal transmission is relatively sensitive to PAPR or CMproperties as compared to downlink signal transmission. The increase ofpower amplifier costs caused by the increase of PAPR or CM propertiesmay generate more serious problems in a user equipment (UE). Thus, theSC-FDMA scheme is used for uplink signal transmission.

FIG. 5 is a conceptual diagram illustrating a general SC-FDMA scheme.

As shown in FIG. 5, the OFDM scheme and the SC-FDMA scheme areconsidered to be identical to some extent with each other, because theyconvert a serial signal into parallel signals, map the parallel signalsto subcarriers, perform an IDFT or IFFT process on the maple signals,convert the IDFT- or IFFT-processed signals into a serial signal, attacha cyclic prefix (CP) to the resultant serial signal, and transmit the CPresultant signal via a radio frequency (RF) module. However, in contrastto the OFDM scheme, the SC-FDMA scheme converts parallel signals into aserial signal, and performs DFT spreading upon the serial signal, sothat it reduces the influence of a next IDFT or IFFT process andmaintains a single signal characteristic of more than a predeterminedlevel as much as possible.

In the meantime, the reason why the CM value is degraded when a MIMOscheme is applied to uplink signal transmission is as follows. If aplurality of single-carrier signals each having good CM properties issimultaneously overlapped with each other, the overlapped signals mayhave poor CM properties. Therefore, if the SC-FDMA system multiplexesoutput information of several layers using a minimum number ofsingle-carrier signals or one single-carrier signal on a single physicalantenna, a transmission signal having a good CM can be generated.

A codeword-layer mapping process may be performed before information tobe transmitted is precoded. Since the SC-FDMA scheme is generally usedfor one transmission mode (1Tx), the number of layers is 1. However, ifthe SC-FDMA scheme supports a MIMO scheme, the number of layers isplural, and a codeword composed of a single transport block may bemapped to a plurality of layers.

FIG. 6 is a conceptual diagram illustrating a method for mapping acodeword to several layers.

Referring to FIG. 6, if the codeword-layer mapping is carried out aftera DFT process for the SC-FDMA scheme is performed, a CM value may beincreased. That is, because an output signal of a DFT block undergoesother processes before entering an IFFT module, i.e., because the outputsignal of the DFT block is divided into two layers, a CM value may beincreased.

FIG. 7 is a conceptual diagram illustrating a method for performing DFTupon each layer after performing codeword-to-layer mapping (i.e., acodeword-layer mapping) so as to prevent a CM value for each antennafrom being increased.

Therefore, if the number of DFT blocks is changed while being classifiedaccording to layer numbers based on a rank value, a low CM value can bemaintained. That is, the output signal of the DFT block is directlyinput to the IFFT block without passing through other processes, so thata low CM value can be maintained. In the case of actual implementation,a plurality of layers may share a single DFT block.

If a plurality of layer signals is transmitted via a single antenna byapplying the MIMO scheme to uplink signal transmission, a PAPR or a CMproperty may be deteriorated. In order to overcome the above-mentionedproblem, the following embodiments of the present invention willdescribe a method for designing a codebook based on a precoding matrixby which only one layer signal is transmitted via a single antenna. Inaddition, the principle of designing a codebook that includes anappropriate number of precoding matrices and maximizes a chordaldistance among precoding matrices of the codebook will hereinafter bedescribed in detail.

Next, a chordal distance will firstly be described, and an uplinkcodebook (specifically, a codebook for use in both 4-antenna and Rank 3transmission) and a method for transmitting and receiving a signal usingthe codebook will hereinafter be described in detail.

One embodiment of the present invention provides a method for generatinga 4Tx-rank 3 codebook using 6 precoding matrices shown in the followingTable 1. The equivalent precoding matrices shown in Table 1 generate thesame Signal to Interference and Noise Ratio (SINR) value given a channelcondition. In the following embodiments, it is assumed that acodeword-to-layer mapping (also called a codeword-layer mapping) for usein rank 3 transmission is carried out as shown in FIG. 3. That is, it isassumed that Codeword 1 is mapped to Layer 1 and Codeword 2 is equallymapped to Layer 2 and Layer 3 in units of a symbol.

TABLE 1 General codebook Group form Equivalent precoding matrices 1$\quad\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & X & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & X & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 0 & X \\1 & 0 & 0 \\0 & 1 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 0 & X \\0 & 1 & 0 \\1 & 0 & 0\end{bmatrix}$ 2 $\quad\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & X & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & X & 0 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & X & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & X & 0 \\1 & 0 & 0\end{bmatrix}$ 3 $\quad\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}$ $\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\X & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\0 & X & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\0 & X & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & X\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & X\end{bmatrix}$ 4 $\quad\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & X & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\0 & X & 0 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 0 & X \\0 & 1 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & X \\1 & 0 & 0\end{bmatrix}$ 5 $\quad\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}$ $\begin{bmatrix}0 & 0 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\X & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & X & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & X & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\0 & 0 & X\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\0 & 0 & X\end{bmatrix}$ 6 $\quad\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}$ $\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\0 & X & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & X & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & X\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\0 & 0 & X\end{bmatrix}$

The precoding matrix for use in actual uplink signal transmission andreception is configured in a multiplication format in which eachprecoding matrix shown in Table 1 is multiplied by a specific constant.In the following description, the constant multiplied by each precodingmatrix will herein be omitted for convenience of description otherwisethe constant is mentioned.

In the meantime, it can be shown that permutation between specificcolumns does not affect SINR performance. For example, although aprecoding matrix [C1, C2, C3] is modified into another precoding matrix[C1, C3, C2], this modification is considered to merely be codewordswapping, and has no change in total sum resultant SNR values. In thefollowing description, only “General Codebook Form” in each of 6precoding matrix groups shown Table 1 will hereinafter be described, butequivalent precoding matrices will not be described.

On the other hand, the codebook shown in Table 1 may include a varietyof precoding matrices according to X values of precoding matrices. Ifthe X value is any one of QPSK alphabets (such as 1, −1, j and −j), thefollowing precoding matrix groups may be obtained represented by thefollowing Table 2.

TABLE 2 Gr Equation Codebook subsets Gr1 $\quad\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ Gr2 $\quad\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Gr3 $\quad\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}$ Gr4 $\quad\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Gr5 $\quad\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}$ Gr6 $\quad\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}$ $\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}$

Table 2 illustrates that an X parameter of each precoding matrix groupis denoted by Xε{±1, ±j}.

The above-mentioned method for using the entirety the precoding matricesas a 4Tx-rank 3 codebook can also be used. One embodiment of the presentinvention provides a method for designing a codebook that includes amaximum chordal distance among precoding matrices contained in thecodebook using the chordal distance concept and at the same timeincludes an optimized number of precoding matrices.

FIG. 8 is conceptual diagram illustrating a chordal distance.

A chordal distance is well known as one of norms (or standards) forcomparing performance of various codebook sets. Herein, the term“chordal” indicates a straight line between two points located at thecircumference. Therefore, given a two-dimensional (2D) case, a chordaldistance indicates a distance between two points located at thecircumference of a circle (e.g., a unit circle) as shown in FIG. 8.Generally, it is preferable that a codebook including precodingmatrices, each of which has a long chordal distance, include a varietyof precoding matrices to be used in different channel conditions.

There is a need for the 4Tx-codebook to consider a four-dimensionalchordal distance, so that the following equation 3 can be used as achordal distance for selecting a codebook set.

$\begin{matrix}{{d_{c}\left( {P,Q} \right)} = {\frac{1}{\sqrt{2}}{{{PP}^{H} - {QQ}^{H}}}_{F}}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

In Equation 3, P is P=[v₁ v₂ . . . v_(N)], and Q is Q=[u₁ u₂ . . .u_(N)], where v_(i) and u_(i) (i=1, 2, . . . N, N=4 in the case of 4Txantennas) are principal vectors of the matrices P and Q, respectively.In addition,

${A}_{F} = {\sqrt{\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n}{a_{ij}}^{2}}} = \sqrt{{trace}\left( {AA}^{H} \right)}}$is the Frobenius norm of the matrix. The above-mentioned chordaldistance can also be measured by the following equation 4.

$\begin{matrix}\begin{matrix}{{d_{c}\left( {P,Q} \right)} = {\frac{1}{\sqrt{2}}{{{PP}^{H} - {QQ}^{H}}}_{F}}} \\{= \sqrt{n - {{trace}\left( {{AA}^{H}{BB}^{H}} \right)}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$where A and B are orthonormal generation magtrices for P and Qrespectively

The above-mentioned codebook design for the 4Tx system based on fourtransmission antennas will be described using the above-mentionedchordal distance concept.

First, the chordal distance among 6 precoding matrix groups can becalculated as follows.

A chordal distance between a precoding matrix X_(i) of the precodingmatrix group Gr_(i) and a precoding matrix X_(j) of the precoding matrixgroup Gr_(j) is dependent upon a trace (X_(i)X_(i) ^(H)−X_(j)X_(j) ^(H))(X_(i)X_(i) ^(H)−X_(j)X_(j) ^(H))^(H). This trace may reflect a distancebetween X_(i)X_(i) ^(H) and X_(j)X_(j) ^(H). The number of cases inwhich two pairs are selected from among the above 6 groups is set to arelatively high number, e.g., 15

$\left( {= \begin{pmatrix}6 \\2\end{pmatrix}} \right),$such that the following description will consider and disclose only adifference between two precoding matrices selected from among differentgroups. First, the following values will be calculated as represented bythe following equation 5.

$\begin{matrix}{{{X_{1}X_{1}^{H}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 \\x_{1} & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & \sqrt{2}\end{bmatrix}} \times {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & x_{1}^{*} & 0 & 0 \\0 & 0 & \sqrt{2} & 0 \\0 & 0 & 0 & \sqrt{2}\end{bmatrix}}} = {\frac{1}{2}\begin{bmatrix}1 & x_{1}^{*} & 0 & 0 \\x_{1} & {x_{1}}^{2} & 0 & 0 \\0 & 0 & 2 & 0 \\0 & 0 & 0 & 2\end{bmatrix}}}}{{X_{2}X_{2}^{H}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 \\0 & \sqrt{2} & 0 \\x_{2} & 0 & 0 \\0 & 0 & \sqrt{2}\end{bmatrix}} \times {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & x_{2}^{*} & 0 \\0 & \sqrt{2} & 0 & 0 \\0 & 0 & 0 & \sqrt{2}\end{bmatrix}}} = {\frac{1}{2}\begin{bmatrix}1 & 0 & x_{2}^{*} & 0 \\0 & 2 & 0 & 0 \\x_{2} & 0 & {x_{2}}^{2} & 0 \\0 & 0 & 0 & 2\end{bmatrix}}}}{{X_{3}X_{3}^{H}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & \sqrt{2} \\x_{3} & 0 & 0\end{bmatrix}} \times {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 & x_{3}^{*} \\0 & \sqrt{2} & 0 & 0 \\0 & 0 & \sqrt{2} & 0\end{bmatrix}}} = {\frac{1}{2}\begin{bmatrix}1 & 0 & 0 & x_{3}^{*} \\0 & 2 & 0 & 0 \\0 & 0 & 2 & 0 \\x_{3} & 0 & 0 & {x_{3}}^{2}\end{bmatrix}}}}{{X_{4}X_{4}^{H}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & \sqrt{2} & 0 \\1 & 0 & 0 \\x_{4} & 0 & 0 \\0 & 0 & \sqrt{2}\end{bmatrix}} \times {\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 & x_{4}^{*} & 0 \\\sqrt{2} & 0 & 0 & 0 \\0 & 0 & 0 & \sqrt{2}\end{bmatrix}}} = {\frac{1}{2}\begin{bmatrix}2 & 0 & 0 & 0 \\0 & 1 & x_{4}^{*} & 0 \\0 & x_{4} & {x_{4}}^{2} & 0 \\0 & 0 & 0 & 2\end{bmatrix}}}}{{X_{5}X_{5}^{H}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & \sqrt{2} & 0 \\1 & 0 & 0 \\0 & 0 & \sqrt{2} \\x_{5} & 0 & 0\end{bmatrix}} \times {\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 & 0 & x_{5}^{*} \\\sqrt{2} & 0 & 0 & 0 \\0 & 0 & \sqrt{2} & 0\end{bmatrix}}} = {\frac{1}{2}\begin{bmatrix}2 & 0 & 0 & 0 \\0 & 1 & 0 & x_{5}^{*} \\0 & 0 & 0 & 0 \\0 & x_{5} & 0 & {x_{5}}^{2}\end{bmatrix}}}}{{X_{6}X_{6}^{H}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & \sqrt{2} & 0 \\1 & 0 & \sqrt{2} \\0 & 0 & 0 \\x_{6} & 0 & 0\end{bmatrix}} \times {\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 0 & 1 & x_{6}^{*} \\\sqrt{2} & 0 & 0 & 0 \\0 & \sqrt{2} & 0 & 0\end{bmatrix}}} = {\frac{1}{2}\begin{bmatrix}2 & 0 & 0 & 0 \\0 & 2 & 0 & 0 \\0 & 0 & 1 & x_{6}^{*} \\0 & 0 & x_{6} & {x_{6}}^{2}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In Equation 5, Xi is a precoding matrix belonging to the group i. Adifference between a precoding matrix of. the group 1 and a precodingmatrix of another group can be represented by the following equation 6.

$\begin{matrix}{\mspace{79mu}{{{{X_{1}X_{1}^{H}} - {X_{2}X_{2}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & x_{1}^{*} & {- x_{2}^{*}} & 0 \\x_{1} & {{x_{1}}^{2} - 2} & 0 & 0 \\{- x_{2}} & 0 & {2 - {x_{2}}^{2}} & 0 \\0 & 0 & 0 & 0\end{bmatrix}}}{{{X_{1}X_{1}^{H}} - {X_{3}X_{3}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & x_{1}^{*} & {- x_{2}^{*}} & {- x_{3}^{*}} \\x_{1} & {{x_{1}}^{2} - 2} & 0 & 0 \\0 & 0 & 0 & 0 \\{- x_{3}} & 0 & 0 & {2 - {x_{3}}^{3}}\end{bmatrix}}}\mspace{79mu}{{{X_{1}X_{1}^{H}} - {X_{4}X_{4}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & x_{1}^{*} & 0 & 0 \\x_{1} & {{x_{1}}^{2} - 1} & {- x_{4}^{*}} & 0 \\0 & {- x_{4}} & {2 - {x_{4}}^{2}} & 0 \\0 & 0 & 0 & 0\end{bmatrix}}}\mspace{79mu}{{{X_{1}X_{1}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & x_{1}^{*} & 0 & 0 \\x_{1} & {{x_{1}}^{2} - 1} & 0 & {- x_{5}^{*}} \\0 & 0 & 0 & 0 \\0 & {- x_{5}} & 0 & {2 - {x_{5}}^{2}}\end{bmatrix}}}\mspace{79mu}{{{X_{1}X_{1}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & x_{1}^{*} & 0 & 0 \\x_{1} & {{x_{1}}^{2} - 2} & 0 & 0 \\0 & 0 & 1 & {- x_{6}^{*}} \\0 & 0 & {- x_{6}} & {2 - {x_{6}}^{2}}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

A difference between a precoding matrix of the group 2 and a precodingmatrix of another group is represented by the following equation 7.

$\begin{matrix}{{{{X_{2}X_{2}^{H}} - {X_{3}X_{3}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & x_{2}^{*} & {- x_{3}^{*}} \\0 & 0 & 0 & 0 \\x_{2} & 0 & {{x_{2}}^{2} - 2} & 0 \\{- x_{3}} & 0 & 0 & {2 - {x_{2}}^{2}}\end{bmatrix}}}{{{X_{2}X_{2}^{H}} - {X_{4}X_{4}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & x_{2}^{*} & 0 \\0 & 1 & {- x_{4}^{*}} & 0 \\x_{2} & {- x_{4}} & {{x_{2}}^{2} - {x_{4}}^{2}} & 0 \\0 & 0 & 0 & 0\end{bmatrix}}}{{{X_{2}X_{2}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & x_{2}^{*} & 0 \\0 & 1 & 0 & {- x_{5}^{*}} \\x_{2} & 0 & {{x_{2}}^{2} - 2} & 0 \\0 & {- x_{5}} & 0 & {2 - {x_{5}}^{2}}\end{bmatrix}}}{{{X_{2}X_{2}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & x_{2}^{*} & 0 \\0 & 0 & 0 & 0 \\x_{2} & 0 & {{x_{2}}^{2} - 1} & {- x_{6}^{*}} \\0 & 0 & {- x_{6}} & {2 - {x_{6}}^{2}}\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

In addition, a difference between a precoding matrix of the group 3 anda precoding matrix of another group is represented by the followingequation 8.

$\begin{matrix}{{{X_{3} X_{3}^{H}} - {X_{4} X_{4}^{H}}} = {\frac{1}{2}\left\lbrack \begin{matrix}{- 1} & 0 & 0 & x_{3}^{*} \\0 & 1 & {- x_{4}^{*}} & 0 \\0 & {- x_{4}} & {2 - {x_{4}}^{2}} & 0 \\x_{3} & 0 & 0 & {{x_{3}}^{2} - 2}\end{matrix} \right\rbrack}} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack \\{{{X_{3}X_{3}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & 0 & x_{3}^{*} \\0 & 1 & 0 & {- x_{5}^{*}} \\0 & 0 & 0 & 0 \\x_{3} & {- x_{5}} & 0 & {{x_{3}}^{2} - {x_{5}}^{2}}\end{bmatrix}}} & \; \\{{{X_{3}X_{3}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & 0 & x_{3}^{*} \\0 & 0 & 0 & 0 \\0 & 0 & 1 & {- x_{6}^{*}} \\x_{3} & 0 & {- x_{6}} & {{x_{3}}^{2} - {x_{6}}^{2}}\end{bmatrix}}} & \;\end{matrix}$

In addition, a difference between a precoding matrix of the group 4 anda precoding matrix of another group is represented by the followingequation 9.

$\begin{matrix}{{{{X_{4}X_{4}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & 0 & 0 \\0 & 0 & x_{4}^{*} & {- x_{5}^{*}} \\0 & x_{4} & {{x_{4}}^{2} - 2} & 0 \\0 & {- x_{5}} & 0 & {2 - {x_{5}}^{2}}\end{bmatrix}}}{{{X_{4}X_{4}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & 0 & 0 \\0 & {- 1} & x_{4}^{*} & 0 \\0 & x_{4} & {{x_{4}}^{2} - 1} & {- x_{6}^{*}} \\0 & 0 & {- x_{6}} & {2 - {x_{6}}^{2}}\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack\end{matrix}$

Finally, a difference between a precoding matrix of the group 5 and aprecoding matrix of the group 6 is represented by the following equation10.

$\begin{matrix}{{{X_{5}X_{5}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & 0 & 0 \\0 & {- 1} & 0 & x_{5}^{*} \\0 & 0 & {- 1} & {- x_{6}^{*}} \\0 & x_{5} & {- x_{6}} & {{x_{5}}^{2} - {x_{6}}^{2}}\end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack\end{matrix}$

It should be noted that variables shown in Equations 6 to 10 are locatedin a unit circle. In addition, Equations 6 to 10 can be simplified asshown in the following equations 11 and 12.

$\begin{matrix}\left\{ \begin{matrix}{{{X_{1}X_{1}^{H}} - {X_{2}X_{2}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & x_{1}^{*} & {- x_{2}^{*}} & 0 \\x_{1} & {- 1} & 0 & 0 \\{- x_{2}} & 0 & 1 & 0 \\0 & 0 & 0 & 0\end{bmatrix}}} \\{{{X_{1}X_{1}^{H}} - {X_{3}X_{3}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & x_{1}^{*} & 0 & {- x_{3}^{*}} \\x_{1} & {- 1} & 0 & 0 \\0 & 0 & 0 & 0 \\{- x_{3}} & 0 & 0 & 1\end{bmatrix}}} \\{{{X_{1}X_{1}^{H}} - {X_{4}X_{4}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & x_{1}^{*} & 0 & 0 \\x_{1} & 0 & {- x_{4}^{*}} & 0 \\0 & {- x_{4}} & 1 & 0 \\0 & 0 & 0 & 0\end{bmatrix}}} \\{{{X_{1}X_{1}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & x_{1}^{*} & 0 & 0 \\x_{1} & 0 & 0 & {- x_{5}^{*}} \\0 & 0 & 0 & 0 \\0 & {- x_{3}} & 0 & 1\end{bmatrix}}}\end{matrix} \right. & \left\lbrack {{Equation}\mspace{14mu} 11} \right\rbrack \\\left\{ \begin{matrix}{{{X_{2}X_{2}^{H}} - {X_{3}X_{3}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & x_{2}^{*} & {- x_{3}^{*}} \\0 & 0 & 0 & 0 \\x_{2} & 0 & {- 1} & 0 \\{- x_{3}} & 0 & 0 & 1\end{bmatrix}}} \\{{{X_{2}X_{2}^{H}} - {X_{4}X_{4}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & x_{2}^{*} & 0 \\0 & 1 & {- x_{4}^{*}} & 0 \\x_{2} & {- x_{4}} & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}}} \\{{{X_{2}X_{2}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & x_{2}^{*} & 0 \\0 & 0 & 0 & 0 \\x_{2} & 0 & 1 & {- x_{6}^{*}} \\0 & 0 & {- x_{6}} & 1\end{bmatrix}}}\end{matrix} \right. & \; \\\left\{ \begin{matrix}{{{X_{3}X_{3}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & 0 & x_{3}^{*} \\0 & 1 & 0 & {- x_{5}^{*}} \\0 & 0 & 0 & 0 \\x_{3} & {- x_{5}} & 0 & 0\end{bmatrix}}} \\{{{X_{3}X_{3}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & 0 & x_{3}^{*} \\0 & 0 & 0 & 0 \\0 & 0 & 1 & {- x_{6}^{*}} \\x_{3} & 0 & {- x_{6}} & 0\end{bmatrix}}}\end{matrix} \right. & \; \\\left\{ \begin{matrix}{{{X_{4}X_{4}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & 0 & x_{3}^{*} \\0 & 0 & x_{4}^{*} & {- x_{5}^{*}} \\0 & x_{4} & {- 1} & 0 \\0 & {- x_{5}} & 0 & 1\end{bmatrix}}} \\{{{X_{4}X_{4}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & 0 & 0 \\0 & {- 1} & x_{4}^{*} & 0 \\0 & x_{4} & 0 & {- x_{6}^{*}} \\0 & 0 & {- x_{6}} & 1\end{bmatrix}}}\end{matrix} \right. & \; \\\left\{ {{{X_{5}X_{5}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & 0 & 0 & 0 \\0 & {- 1} & 0 & x_{5}^{*} \\0 & 0 & 1 & {- x_{6}^{*}} \\0 & x_{5} & {- x_{6}} & 1\end{bmatrix}}} \right. & \; \\{{{{X_{1}X_{1}^{H}} - {X_{6}X_{6}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & x_{1}^{*} & 0 & 0 \\x_{1} & {- 1} & 0 & 0 \\0 & 0 & 1 & {- x_{6}^{*}} \\0 & 0 & {- x_{6}} & 1\end{bmatrix}}}{{{X_{2}X_{2}^{H}} - {X_{5}X_{5}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & x_{2}^{*} & 0 \\0 & 1 & 0 & {- x_{5}^{*}} \\x_{2} & 0 & {- 1} & 0 \\0 & {- x_{5}} & 0 & 1\end{bmatrix}}}{{{X_{3}X_{3}^{H}} - {X_{4}X_{4}^{H}}} = {\frac{1}{2}\begin{bmatrix}{- 1} & 0 & 0 & x_{3}^{*} \\0 & 1 & {- x_{4}^{*}} & 0 \\0 & {- x_{4}} & {- 1} & 0 \\x_{3} & 0 & 0 & {- 1}\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack\end{matrix}$

The numbers at zero values in a difference X_(i)X_(i) ^(H)−X_(j)X_(j)^(H) between different matrices of Equation 11 are the same when(i,j)≠(1,6), (6,1), (2,5), (5,2), (3,4) or (4,3), but the individualzero values have different positions. In more detail, under the case of(i,j)≠(1,6), (6,1), (2,5), (5,2), (3,4) or (4,3), a difference matrixincludes elements (+1, −1, x_(i), x_(i) ^(o), −x_(j) and −x_(j) ^(o)),each of which is not zero. Therefore, a trace of the difference matrixX_(i)X_(i) ^(H)−X_(j)X_(j) ^(H) becomes

$\frac{\sqrt{3}}{2}$as represented by the following equation 13.

                                     [Equation  13] $\begin{matrix}{{d_{c}\left( {X_{i},X_{j}} \right)} = {\frac{1}{\sqrt{2}}{{{X_{i}X_{i}^{H}} - {X_{j}X_{j}^{H}}}}_{F}}} \\{= {\frac{1}{\sqrt{2}}\sqrt{{trace}\left( {\left( {{X_{i}X_{i}^{H}} - {H_{j}X_{j}^{H}}} \right)\left( {{X_{i}X_{i}^{H}} - {X_{j}X_{j}^{H}}} \right)^{H}} \right)}}} \\{= {\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{1}{4} \cdot \left( {{1}^{2} + {{- 1}}^{2} + {x_{i}}^{2} + {x_{i}^{*}}^{2} + {{- x_{j}}}^{2} + {{- x_{j}^{*}}}^{2}} \right)}}} \\{= {\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{1}{4}\left( {1 + 1 + 1 + 1 + 1 + 1} \right)}}} \\{= \frac{\sqrt{3}}{2}}\end{matrix}$   where$\mspace{20mu}{x_{i},{x_{j} \in \left\{ {{\mathbb{e}}^{j\frac{2\pi}{N} \times 0},{\mathbb{e}}^{j\frac{2\pi}{N} \times 1},\ldots\mspace{14mu},{\mathbb{e}}^{j\frac{2\pi}{N} \times {({N - 1})}}} \right\}},\mspace{20mu}{N = 2^{1}},2^{2},2^{3},2^{4},{{\ldots\mspace{20mu}\left( {i,j} \right)} \neq \left( {1,6} \right)},\left( {6,1} \right),\left( {2,5} \right),\left( {5,2} \right),{\left( {3,4} \right)\mspace{14mu}{or}\mspace{14mu}\left( {4,3} \right)}}$

On the other hand, the numbers of zero values in a difference X_(i)X_(i)^(H)−X_(j)X_(j) ^(H) between different matrices of Equation 12 are thesame when (i,j)=(1,6), (6,1), (2,5), (5,2), (3,4) or (4,3), but theindividual 0 values have different positions. In more detail, under thecase of (i,j)=(1,6), (6,1), (2,5), (5,2), (3,4) or (4,3), a differencematrix includes elements (+1, +1, −1, −1, x_(i), x_(i) ^(o), −x_(j) and−x_(j) ^(o)), each of which is not identical to zero. Therefore, a traceof the difference matrix X_(i)X_(i) ^(H)−X_(j)X_(j) ^(H) becomes 1 asrepresented by the following equation 14.

                                [Equation  14] $\begin{matrix}{{d_{c}\left( {X_{i},X_{j}} \right)} = {\frac{1}{\sqrt{2}}{{{X_{i}X_{i}^{H}} - {X_{j}X_{j}^{H}}}}_{F}}} \\{= {\frac{1}{\sqrt{2}}\sqrt{{trace}\left( {\left( {{X_{i}X_{i}^{H}} - {H_{j}X_{j}^{H}}} \right)\left( {{X_{i}X_{i}^{H}} - {X_{j}X_{j}^{H}}} \right)^{H}} \right)}}} \\{= {\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{1}{4} \cdot \begin{pmatrix}{{2 \times {1}^{2}} + {2 \times {{- 1}}^{2}} + {x_{i}}^{2} +} \\{{x_{i}^{*}}^{2} + {{- x_{j}}}^{2} + {{- x_{j}^{*}}}^{2}}\end{pmatrix}}}} \\{= {\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{1}{4}\left( {2 + 2 + 1 + 1 + 1 + 1} \right)}}} \\{= 1}\end{matrix}$ where$x_{i},{x_{j} \in \left\{ {{\mathbb{e}}^{j\frac{2\pi}{N} \times 0},{\mathbb{e}}^{j\frac{2\pi}{N} \times 1},\ldots\mspace{14mu},{\mathbb{e}}^{j\frac{2\pi}{N} \times {({N - 1})}}} \right\}},{N = 2^{1}},2^{2},2^{3},2^{4},{{\ldots\left( {i,j} \right)} \neq \left( {1,6} \right)},\left( {6,1} \right),\left( {2,5} \right),\left( {5,2} \right),{\left( {3,4} \right)\mspace{14mu}{or}\mspace{14mu}\left( {4,3} \right)}$

If the pair of groups shown in Equations 13 and 14 satisfies(i,j)=(1,6), (6,1), (2,5), (5,2), (3,4) or (4,3), a chordal distance of1 is provided; otherwise, a chordal distance of

$\frac{\sqrt{3}}{2}$is provided.

FIG. 9 illustrates the relationship of chordal distances among 6precoding matrix groups according to one embodiment of the presentinvention.

The following theorem can be derived using the relationship shown inFIG. 9.

<Theorem 1>

If a pair of groups shown in Table 2 satisfies (i,j)=(1,6), (6,1),(2,5), (5,2), (3,4) or (4,3), a chordal distance of 1 is provided. If apair of groups shown in Table 2 satisfies (i,j)≠(1,6), (6,1), (2,5),(5,2), (3,4) or (4,3), a chordal distance of

$\frac{\sqrt{3}}{2}$is provided.

On the other hand, the relationship of chordal distances among precodingmatrices of the precoding matrix group shown in Table 2 will hereinafterbe described in detail.

First, a chordal distance among different precoding matrices containedin same precoding matrix group, i.e. group 1 (Gr 1) can be calculated asshown in the following Equation 15.

                                [Equation  15] $\begin{matrix}{{d_{c}\left( {X_{1},X_{1}} \right)} = {\frac{1}{\sqrt{2}}{{{X_{1}X_{1}^{H}} - {Y_{1}Y_{1}^{H}}}}_{F}}} \\{= {\frac{1}{\sqrt{2}}\sqrt{{trace}\left( {\left( {{X_{1}X_{1}^{H}} - {Y_{1}Y_{1}^{H}}} \right)\left( {{X_{1}X_{1}^{H}} - {Y_{1}Y_{1}^{H}}} \right)^{H}} \right)}}} \\{= {\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{1}{4} \cdot \left\lbrack {{2{{x_{1} - y_{1}}}^{2}} + \left( {{x_{1}}^{2} - {y_{1}}^{2}} \right)} \right\rbrack}}} \\{= {\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{1}{4} \cdot \left( {2{{x_{1} - y_{1}}}^{2}} \right)}}} \\{= {\frac{1}{2} \cdot \sqrt{{{x_{1} - y_{1}}}^{2}}}}\end{matrix}$ where ${X_{1} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 \\x_{1} & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & \sqrt{2}\end{bmatrix}} \in {{Gr}\; 1}}},{Y_{1} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 \\y_{1} & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & \sqrt{2}\end{bmatrix}} \in {{Gr}\; 1}}},{{{X_{1}X_{1}^{H}} - {Y_{1}Y_{1}^{H}}} = {\frac{1}{2}\begin{bmatrix}0 & {x_{1}^{*} - y_{1}^{*}} & 0 & 0 \\{x_{1} - y_{1}} & {{x_{1}}^{2} - {y_{1}}^{2}} & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}}}$$x_{1},{y_{1} \in \left\{ {{\mathbb{e}}^{j\frac{2\pi}{N} \times 0},{\mathbb{e}}^{j\frac{2\pi}{N} \times 1},\ldots\mspace{14mu},{\mathbb{e}}^{j\frac{2\pi}{N} \times {({N - 1})}}} \right\}},{{{where}\mspace{14mu} N} = 2^{1}},2^{2},2^{3},2^{4},\ldots$

As can be seen from Equation 15, a chordal distance among differentprecoding matrices contained in the Group 1 (Gr 1) is proportional tothe distance among variables of individual precoding matrices.Wherefore, as can be seen from Equation 16; a maximum chordal distancewithin the same group may be ‘1’ as represented by the followingequation 16.

$\begin{matrix}{\begin{matrix}{{d_{c}\left( {X_{1},Y_{1}} \right)} = {\frac{1}{\sqrt{2}}{{{X_{1}X_{1}^{H}} - {Y_{1}Y_{1}^{H}}}}_{F}}} \\{= {\frac{1}{2} \cdot \sqrt{{{x_{1} - y_{1}}}^{2}}}} \\{= {\frac{1}{2}\sqrt{{{{\mathbb{e}}^{{j\theta}_{x_{1}}} - {\mathbb{e}}^{{j\theta}_{y_{1}}}}}^{2}}}} \\{= {\frac{1}{2}\sqrt{{{\mathbb{e}}^{{j\theta}_{x_{1}}}}^{2}{{1 - {\mathbb{e}}^{{j\theta}_{1}}}}^{2}}}} \\{= {{\frac{1}{2}\sqrt{{{1 - {\mathbb{e}}^{{j\theta}_{1}}}}^{2}}} \leq {\frac{1}{2}{\sqrt{4} = 1}}}}\end{matrix}{{{{where}\mspace{14mu} x_{1}} = {\mathbb{e}}^{{j\theta}_{x_{1}}}},{y_{1} = {\mathbb{e}}^{{j\theta}_{y_{1}}}},{\theta_{1} = {\theta_{x_{1}} - \theta_{y_{1}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$

In order to implement the maximum chordal distance, it is necessary forvariables contained in each precoding matrix to satisfy the relationshipshown in FIG. 10.

FIG. 10 illustrates conditions of variables for providing a maximumchordal distance within the same precoding matrix group.

In other words, a precoding matrix including two variables having aphase difference of 180° therebetween may have a chordal distance of 1.The above-mentioned principle may also be equally applied to theremaining groups other than the group 1 (Gr 1), and the followingtheorem 2 may be derived.

<Theorem 2>

In Table 2, as to a chordal distance bet teen two precoding matricescontained in the same precoding matrix group, if variables of theprecoding matrices have a phase difference of 180° therebetween, i.e.,if variables of the precoding matrices are located at a diameter of aunit circle, the chordal distance of 1 can be obtained.

<Proof of Theorem 2>

As can be seen from Equation 16, it can be recognized that theorem 2achieved in the first group 1 (Gr 1), and the same conclusion may alsobe derived from other groups.

As an application of theorem 2, a codebook subset for satisfying theorem2 in the precoding matrix group 1 can be designed as represented by thefollowing Table 3.

TABLE 3 Codebook Subset $\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$

In Table 3, a basic precoding matrix is denoted by

$\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},$and a codebook subset for satisfying the relationship between the basicprecoding matrix and theorem 2 is illustrated. However, it should benoted that a variety of subsets may be designed according to methods forselecting such a basic precoding matrix using the same principles asdescribed above. In this case, a codebook for implementing the maximumchordal distance may include an excessively small number of precodingmatrices as necessary. Therefore, the following description assumes thatvariables contained in each precoding matrix group are denoted by QPSKalphabets, such that the chordal distance relationship can berepresented by the following equation 17.

$\begin{matrix}\begin{matrix}{{d_{c}\left( {X_{1},Y_{1}} \right)} = {\frac{1}{\sqrt{2}}{{{X_{1}X_{1}^{H}} - {Y_{1}Y_{1}^{H}}}}_{F}}} \\{= {\frac{1}{2} \cdot \sqrt{{{x_{1} - y_{1}}}^{2}}}} \\{= {\frac{1}{2}\sqrt{{{1 - {\mathbb{e}}^{{j\theta}_{1}}}}^{2}}}} \\{= \left\{ \begin{matrix}{{1\mspace{14mu}\ldots\mspace{14mu}\theta_{1}} = \pi} \\{{\frac{1}{\sqrt{2}}\mspace{14mu}\ldots\mspace{14mu}\theta_{1}} = {\pm \frac{\pi}{2}}}\end{matrix} \right.}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack\end{matrix}$

FIG. 11 illustrates the relationship of chordal distances when avariable of a precoding matrix contained it the same precoding matrixgroup includes a QPSK alphabet.

The following theorem 3 can be derived from Equation 17 and FIG. 11.

<Theorem 3>

A chordal distance between two precoding matrices, each of which has apair of variables (i.e., (1, −1), (−1, 1), (j, −j) or (−j, j)) withinthe same precoding matrix group, is denoted by 1, a chordal distancebetween two precoding matrices, each of which has a pair of variables(i.e., (1, ±j), (−1, ±j) (j, ±1) or (−j, ±1)) within the same precodingmatrix group, is denoted by

$\frac{1}{\sqrt{2}}.$<Proof of Theorem 3>

Theorem 3 may be demonstrated through Equation 17.

As can be seen from theorems 1 to 3, an available chordal distancebetween precoding matrices shown in Table 2 may be any of 1,

$\frac{\sqrt{3}}{2}\mspace{14mu}{and}\mspace{14mu}{\frac{1}{\sqrt{2}}.}$The 4Tx rank 3 codebook according to one embodiment of the presentinvention will hereinafter be described in detail.<Case of Maximized Chordal Distance>

In accordance with this embodiment of the present invention, consideringthat a maximum chordal distance between two precoding matrices is set to1 on the basis of the above-mentioned theorems 1 to 3, a method fordesigning a codebook having only precoding matrices having a chordaldistance of 1 therebetween will hereinafter be described in detail.

Based on theorem 1, two precoding matrix groups satisfying a chordaldistance of 1 can be selected from among 6 precoding matrix groups. Asone embodiment of the present invention, the first group 1 (Gr 1) andthe sixth group 6 (Gr 6) shown in Table 1 or 2 can be selected. Inaddition, the precoding matrices, which have a chordal distance of 1within the same group according to theorem 2 or 3, correspond to onlytwo precoding matrices, variables of which have a phase difference of180° therebetween. Accordingly, it is assumed that two precodingmatrices, variables of which have the relationship denoted by (1, −1),are selected from each group. As a result, the generated codebook isrepresented by the following table 4.

TABLE 4 Codebook subset $\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$<Corollary 1>

A codebook, selected from table 2, that satisfies a chordal distance of1 between two precoding matrices includes 4 precoding matrices.

<Case in Which Both Chordal Distance and the Number Precoding Matricesare Considered>

As can be seen from Corollary 1, in the case where the codebook iscomprised of only precoding matrices having a maximum chordal distanceof 1 in consideration of only the chordal distance, only four precodingmatrices can be obtained, and thus it is difficult to reflect a varietyof channel statuses using only the four precoding matrices. Therefore, aspecific case in which a second highest chordal distance value of

$\frac{\sqrt{3}}{2}$is implemented is also reflected in one embodiment of the presentinvention in such a manner that the resultant codebook can be designedaccording to the reflected result. For example, in accordance with thisembodiment of the present invention, two precoding matrices (e.g.,precoding matrices having 1 and −1 as variables), each of which has amaximum chordal distance in each group, are selected from each of 6groups shown in Table 1 using theorems 2 and 3, such that the 4Tx-rank 3codebook can be given as shown in the following Table 5.

TABLE 5 Codebook subset $\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$

Table 5 illustrates a codebook subset. In more detail, Table 5illustrates concepts of subsets related to rank 3 among codebooks of allranks in the 4-antenna system including 4 antennas. From the viewpointof rank 3, a codebook including 12 precoding matrices can be achieved.

In accordance with one embodiment of the present invention, of UserEquipment (UE) selects a specific precoding matrix (in which apredetermined constant for power control is multiplied by the specificprecoding matrix) from among precoding matrices of the rank 3 codebookshown in Table 5, such that the UE can perform precoding and transmitsignals using the selected precoding matrix.

On the other hand, a user equipment (UE) for transmitting uplink signalsusing the above-mentioned codebook and a Node B for receiving the uplinksignals from the UE will hereinafter be described in detail.

FIG. 12 is a block diagram illustrating a general Node B and a generaluser equipment (UE).

Referring to FIG. 12, a Node B 10 includes a processor 11, a memory 12,and a Radio Frequency (RF) unit 13. The RF unit 13 is used as atransmission/reception module for receiving an uplink signal andtransmitting a downlink signal. The processor 11 may control downlinksignal transmission using downlink signal transmission information (forexample, a specific precoding matrix contained in a codebook fordownlink signal transmission) stored in the memory 12. Otherwise, as aninverse process of the precoding process, the processor 11 may control asignal reception process by multiplying uplink signal receptioninformation (e.g., an uplink signal) stored in the memory 12 by aHermitian matrix of the same precoding matrix as a precoding matrix usedin the UE 20.

The UE 20 may include a processor 21, a memory 22, and an RF unit 23used as a transmission/reception module for transmitting an uplinksignal and receiving a downlink signal. The processor 21 may controluplink signal transmission using uplink signal transmission information(for example, a specific precoding matrix contained in theabove-mentioned codebook for uplink signal transmission) stored in thememory 22. Otherwise, as an inverse process of the precoding process,the processor 21 may control a signal reception process by multiplyingdownlink signal reception information (e.g., a downlink signal) storedin the memory 22 by a Hermitian matrix of the same precoding matrix as aprecoding matrix used in the UE 20.

In the meantime, a detailed description about a processor of the UE 20(or the BS 10), particularly, a structure for transmitting a signalusing the SC-FDMA scheme, will hereinafter be described. A processor fortransmitting a signal based on the SC-FDMA scheme in the 3GPP LTE systemand a processor for transmitting a signal based on an OFDM scheme in the3GPP LTE system will hereinafter be described, and a processor forenabling a UE to transmit an uplink signal using the SC-FDMA scheme aswell as the MIMO scheme will then be described below.

FIGS. 13 to 15 illustrate an SC-FDMA scheme for transmitting an uplinksignal in the 3GPP LTE system and an OFDMA scheme for transmitting adownlink signal in the 3GPP LTE system.

Referring to FIG. 13, not only a UE for transmitting an uplink signalbut also a Node B for transmitting a downlink signal includes aSerial-to-Parallel converter 401, a subcarrier mapper 403, an M-pointIDFT module 404, a Parallel-to-Serial converter 405, and the like.However, a UE for transmitting a signal using the SC-FDMA scheme furtherincludes an N-point DFT module 402, and compensates for a predeterminedpart of the IDFT processing influence of the M-point IDFT module 404 sothat a transmission signal can have single carrier characteristics.

FIG. 14 shows the relationship between a block diagram for an uplinksignal process prescribed in TS 36.211 including the 3GPP LTE systemspecification and a processor for transmitting a signal using theSC-FDMA scheme. In accordance with TS 36.211, each UE scrambles atransmission signal using a specific scrambling sequence so as totransmit an uplink signal, and the scrambled signal is modulated so thatcomplex symbols are generated. After that, transform precoding forperforming a DFT spreading process on complex symbols is carried out,That is, a transform precoder prescribed in TS 36.211 may correspond toan N-point DFT module. Thereafter, the DFT-spread signal may be mappedto a specific resource element according to a resource block (RB)-basedmapping rule by a resource element mapper, and it can be recognized thatthis operation corresponds to the subcarrier mapper shown in FIG. 13.The signal mapped to the resource element is M-point IDFT orIFFT—processed by the SC-FDMA signal generator, parallel-to-serialconversion is performed on the IDFT or IFFT processed result, and then acyclic prefix (CP) is added to the P/S conversion result.

In the meantime, FIG. 14 further shows a processor of a Node B that isused to receive a signal which has been received in the base stationthrough the above-mentioned processes.

In this way, the processor for SC-FDMA transmission in the 3GPP LTEsystem does not include a structure for utilizing the MIMO scheme.Therefore, the BS processor for MIMO transmission in the 3GPP LTE systemwill be described first, and a processor for transmitting an uplinksignal by combining the SC-FDMA scheme with the MIMO scheme using theabove BS processor will then be described.

FIG. 15 is a block diagram illustrating a processor for enabling theNode B to transmit a downlink signal using the MIMO scheme in the 3GPPLTE system.

A Node B in the 3GPP LTB system can transmit one or more codewords via adownlink. Therefore, one or more codewords may be processed as complexsymbols by the scrambling module 301 and the modulation mapper 302 inthe same manner as in the uplink operation shown in FIG. 12. Thereafter,the complex symbols are mapped to a plurality of layers by the layermapper 303, and each layer is multiplied by a predetermined precodingmatrix selected according to the channel status and is then allocated toeach transmission antenna by the precoding module 304. The processedtransmission signals of individual antennas are mapped to time-frequencyresource elements to be used for data transmission by the resourceelement mapper 305. Thereafter, the mapped result may be transmitted viaeach antenna after passing through the OFDM signal generator 306.

However, if a downlink signal scheme shown in FIG. 15 is used in the3GPP LTE system, PAPR or CM properties may be degraded. Thus, it isnecessary for a UE to effectively combine the SC-FDMA scheme formaintaining good PAPR and CM properties described in FIGS. 13 and 14with the MIMO scheme shown in FIG. 15, and a UE for performing precodingusing the precoding matrix capable of maintaining good PAPR and CMproperties described in the above embodiment must be constructed.

In accordance with one embodiment of the present invention, it isassumed that a UE for transmitting an uplink signal via multipleantennas (multi-antenna) includes multiple antennas (not shown) fortransmitting and receiving signals. Referring to FIG. 12, the UE 20includes a memory 22 for storing a codebook, and a processor 21 that areconnected to multiple antennas (not shown) and the memory 22 so as toprocess uplink signal transmission. In this case, the codebook stored inthe memory 22 includes precoding matrices shown in Table 5. Theprocessor 21 of the UE configured as described above will hereinafter bedescribed in detail.

FIG. 16 illustrates a processor of the UE according to one embodiment ofthe present invention.

Referring to FIG. 16, the processor of the UE 20 according to oneembodiment of the present invention includes a codeword to layer mapper1401 for mapping uplink signals to a predetermined number of layerscorresponding to a specific rank, a predetermined number of DFT modules1402 for performing Discrete Fourier Transform (DFT) spreading on eachof the predetermined number of layer signals, and a precoder 1403. Theprecoder 1403 selects a specific precoding matrix from a codebookcomprised of 12 precoding matrices obtained when two precoding matricesincluding variables that have a phase difference of 180° therebetweenare selected from each of 6 precoding matrix groups shown in Table 5,thereby precoding a DFT-spread resultant layer signal received from theOFT module 1402. Particularly, in this embodiment of the presentinvention, each DFT module 1402 performs spreading of each layer signal,this DFT module 1402 for spreading each layer signal is located justbefore the precoder 1403. When the precoder 1403 performs precoding, theprecoder 1403 is configured such that each layer signal is mapped to oneantenna and then transmitted via the mapped antenna, so that singlecarrier characteristics of each layer signal are maintained and goodPAPR and CM properties are also maintained. In the meantime, the UE 20further includes a transmission module. The transmission module performsa process constructing an SC-FDMA symbol upon the precoded signal, andtransmits the resultant precoded signal to the Node B via multipleantennas 1405.

In the meantime, the precoder 1403 selects a precoding matrix to be usedfor signal transmission from the codebook stored in the memory 22, andperforms precoding on the selected precoding matrix. Preferably, theseprecoding matrices may be precoding matrices established for equalizingtransmission powers of multiple antennas and/or transmission powers ofrespective layers.

It is assumed that the number of multiple antennas 1405 is 4. Theprocessor of the UE according to one embodiment of the present inventionmay further perform not only a layer shift function for periodically oraperiodically changing a layer mapped to a specific codeword but also anantenna shift function for periodically or aperiodically changing anantenna via which a specific layer signal is transmitted. The layershift function may be performed by the layer mapper 1401 separately fromthe precoding of the precoder 1403, or may also be performed throughcolumn permutation of the precoding matrix when the precoder 1403performs precoding. In addition, the antenna shift function may also becarried out separately from the precoding of the precoder 1403, or mayalso be performed through row permutation of the precoding matrix.

The exemplary embodiments described hereinabove are combinations ofelements and features of the present invention. The elements or featuresmay be considered selective unless otherwise mentioned. Each element orfeature may be practiced without being combined with other elements orfeatures. Further, the embodiments of the present invention may beconstructed by combining parts of the elements and/or features.Operation orders described in the embodiments of the present inventionmay be rearranged. Some constructions or characteristics of any oneembodiment may be included in another embodiment and may be replacedwith corresponding constructions or characteristics of anotherembodiment. It is apparent that the present invention may be embodied bya combination of claims which do not have an explicit cited relation inthe appended claims or may include new claims by amendment afterapplication.

The embodiments of the present invention may be achieved by variousmeans, for example, hardware, firmware, software, or a combinationthereof. In a hardware configuration, the embodiments of the presentinvention may be implemented by one or more application specificintegrated circuits (ASICs), digital signal processors (DSPs), digitalsignal processing devices (DSPDs), programmable logic devices (PLDs),field programmable gate arrays (FPGAs), processors, controllers,microcontrollers, microprocessors, etc.

In a firmware or software configuration, the embodiments of the presentinvention may be achieved by a module, a procedure a function, etc.performing the above-described functions or operations. Software codemay be stored in a memory unit and driven by a processor. The memoryunit may be located at the interior or exterior of the processor and maytransmit data to and receive data from the processor via various knownmeans.

It will be apparent to those skilled in the art that variousmodifications and variations can be made in the present inventionwithout departing from the spirit or scope of the invention. Therefore,the above-mentioned detailed description must be considered only forillustrative purposes instead of restrictive purposes. The scope of thepresent invention must be decided by a rational analysis of claims, andall modifications within equivalent ranges of the present invention arecontained in the scope of the present invention. It is apparent that thepresent invention may be embodied by a combination of claims which donet have an explicit cited relation in the appended claims or mayinclude new claims by amendment after application.

As apparent from the above description, a method for transmitting andreceiving uplink signals according to the embodiments of the presentinvention may transmit and receive using the above-mentioned optimized4Tx Rank 3 codebook, transmit uplink signals by reflecting a channelsituation using a small amount of signaling overhead in various ways,and solve a PAPR problem.

It will be apparent to those skilled in the art that variousmodifications and variations can be made in the present inventionwithout departing from the spirit or scope of the inventions. Thus, itis intended that the present invention covers the modifications andvariations of this invention provided they come within the scope of theappended claims and their equivalents.

What is claimed is:
 1. A method for controlling a user equipment (UE) to transmit uplink signals via four antennas, the method comprising: mapping symbols for the uplink signal to a plurality of layers; performing Discrete Fourier Transform (DFT) spreading upon each of the symbols of the plurality of layers; precoding the DFT-spread symbols of the plurality of layers using a specific precoding matrix selected from a codebook; generating a Single Carrier-Frequency Division Multiple Access (SC-FDMA) symbol based on the precoded symbols; and transmitting the SC-FDMA symbol to a base station (BS) via the four antennas, wherein the codebook consists of precoding matrix groups of $\begin{bmatrix} 1 & 0 & 0 \\ X & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ X & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ X & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ X & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ X & 0 & 0 \end{bmatrix},{{{and}\mspace{14mu}\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ X & 0 & 0 \end{bmatrix}}.}$
 2. The method according to claim 1, wherein each of the precoding matrix groups is satisfied with a condition of Xε{1, −1}.
 3. The method according to claim 1, wherein the codebook consists of precoding matrices of $\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ {- 1} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ {- 1} & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ {- 1} & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ {- 1} & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ {- 1} & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ {- 1} & 0 & 0 \end{bmatrix},$ to each of which a predetermined constant is multiplied.
 4. A method for receiving uplink signal by a base station from a user equipment (UE) configured to use multiple antennas, the method comprising: receiving an uplink signals from the user equipment via the multiple antennas; and performing reverse processing of precoding on the uplink signal using a hermitian matrix corresponding to the specific precoding matrix selected from a codebook, wherein the codebook consists of precoding matrix groups of $\begin{bmatrix} 1 & 0 & 0 \\ X & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ X & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ X & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ X & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ X & 0 & 0 \end{bmatrix},{{and}\mspace{14mu}\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ X & 0 & 0 \end{bmatrix}},$ when symbols for the uplink signals are mapped to three layers and the number of the multiple antennas is four.
 5. The method according to claim 4, wherein the codebook is set to equalize transmission power of multiple antennas.
 6. The method according to claim 4, wherein each of the precoding matrix groups is satisfied with a condition of Xε{1, −1}.
 7. The method according to claim 4, wherein the codebook consists of precoding matrices of $\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ {- 1} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ {- 1} & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ {- 1} & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ {- 1} & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ {- 1} & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ {- 1} & 0 & 0 \end{bmatrix},$ to each of which a predetermined constant is multiplied. 